An operational amplifier has differential gain A_d = 50,000 and common-mode gain A_cm = 2. What is the common-mode rejection ratio (CMRR) expressed in decibels (dB)?

Introduction:
Common-mode rejection ratio (CMRR) quantifies how well a differential amplifier suppresses signals that are common to both inputs relative to signals that are different between them. A high CMRR is critical for rejecting power-line hum, reference shifts, and electromagnetic interference in sensor front-ends and instrumentation systems.

Given Data / Assumptions:

• Differential gain A_d = 50,000.
• Common-mode gain A_cm = 2.
• Standard definition of CMRR, and conversion to decibels using 20 * log10(ratio).

Concept / Approach:
CMRR (ratio) is defined as A_d / A_cm. To express it in decibels, use CMRR_dB = 20 * log10(A_d / A_cm). A larger ratio means the amplifier strongly prefers differential signals and rejects common disturbances.

Step-by-Step Solution:
Compute ratio: CMRR = A_d / A_cm = 50,000 / 2 = 25,000Convert to dB: CMRR_dB = 20 * log10(25,000)Evaluate log10(25,000) ≈ 4.39794CMRR_dB ≈ 20 * 4.39794 ≈ 87.9588 dBRounded to one decimal place: 87.9 dB

Verification / Alternative check:
As a quick estimate, 20 * log10(10,000) = 80 dB and 20 * log10(2.5) ≈ 8 dB; total ≈ 88 dB, consistent with the precise calculation. The resulting value is typical for general-purpose op-amps and below that of precision instrumentation amplifiers.

Why Other Options Are Wrong:

• −87.9 dB / −43.9 dB: CMRR is a suppression ratio and is conventionally reported as a positive dB magnitude.
• 43.9 dB: Would correspond to a ratio of roughly 155, far below the computed 25,000.
• 25,000 (unitless ratio): This is the raw ratio, not the requested dB value.

Common Pitfalls:

• Using 10 * log10 instead of 20 * log10 for voltage or gain ratios.
• Confusing CMRR with PSRR (power-supply rejection ratio).
• Reporting the ratio without specifying units or scale, which can mislead design decisions.

Final Answer:
87.9 dB